Suppose we have a spacecraft in the interstellar medium and we start transmitting a directional beam of ELF radio waves at a frequency -- say, 1 kHz -- that's below the local plasma frequency. Since the beam carries momentum, doing this creates a small force on our spacecraft. Then, when the plasma reflects the beam back at our antenna, we can reflect the radio waves back again, pushing ourselves forward a little bit more. In principle, we could continue reusing the photons to provide even more thrust.

Would this effect let us create a photon rocket that beats the limits set by the (relativistic) rocket equation?

Another way of looking at this idea: naive photon rockets use the energy of a photon very "inefficiently." Suppose we build a nuclear photonic rocket that uses a fission reactor to power a laser beam. Let's say that the photons in the laser beam carry 1 eV each. In Earth's frame of reference, if the rocket is traveling at much less than the speed of light, the Doppler effect is small and those photons still appear to be carrying nearly 1 eV. But if we could recycle the photons, the Doppler effect would transfer some energy to the spacecraft each time the photons are reflected. Thus, the rocket could extract a much larger fraction of the energy in the laser beam.

For the sake of discussion, I'll try to be a bit more specific about what such a device might look like in practice.

The plasma density of the interstellar medium near the Solar System is thought to be about 0.1 cm$^{-3}$, which gives a plasma frequency of

(Applying some common sense to this, it's hard to believe that there are enough electrons in a region of this depth to actually reflect a high-power radio wave. I imagine that radiation pressure would clear out a substantial cavity in the plasma behind the spacecraft. However, the cavity walls should still be reflective.)

Let's say the mass of our spacecraft is $10^{6}$ kg and it has a 1 terawatt fusion reactor on board that converts deuterium and tritium into helium-4, extracting an average of 1 MeV from each reaction. If we discard the reaction products and use the reactor to power a laser beam, the spacecraft gains a momentum of $E/c = 0.0033 \frac{\text{eV} \cdot \text{s}}{\text{m}}$ per reaction while losing 5 amu of mass. For the purpose of the rocket equation, that gives an effective exhaust velocity of $1.3 \times 10^5$ m/s = $0.004c$ and a thrust of 3300 N. But if instead we use the reactor to power an ELF transmitter and, on average, we can get each photon to bounce back once from the interstellar medium and reflect off the spacecraft antenna, then we've just tripled both the thrust and the specific impulse. That brings us quite a bit closer to an interstellar-capable engine. So my question is, would this actually work from a physics perspective?

For the purposes of this question, I'm not interested in practical issues. Assume that we have the technical capability to build an absurdly lightweight superconducting dish antenna thousands of kilometers in diameter, a fusion reactor that can generate ridiculous amounts of power, and radiators that can reject arbitrary amounts of waste heat.

$\begingroup$"Then, when the plasma reflects the beam back at our antenna, we can reflect the radio waves back again, pushing ourselves forward a little bit more. In principle, we could continue reusing the photons to provide even more thrust." This is a misstatement. The reflection will be a (very? )small proportion of the radiated beam, a directional antenna will be missed most of the time by the reflected wave.$\endgroup$
– anna vMay 8 '18 at 5:51

$\begingroup$In the above low-density example, assuming engineering is adequate, I would suspect the greatest limiting factor to be the distribution of scattering angles due to Thomson scattering which would depend on polarization angle.$\endgroup$
– Mathews24May 8 '18 at 5:53

$\begingroup$@annav Why exactly do you think the reflected wave would miss the antenna? It seems to me that because of the large wavelength, the effective aperture of the antenna would be large compared to the distance the radio waves could travel before being reflected. Thus, there's nowhere for the reflected beam to go besides the antenna.$\endgroup$
– ThorondorMay 8 '18 at 6:02

$\begingroup$It would be like covering a radio dish with a giant metal plate.$\endgroup$
– ThorondorMay 8 '18 at 6:06

1 Answer
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If the antenna is very directional and much larger than the waves and before the uniform plasma density distorts it will be like pointing a laser from the center of a mirrored hollow ball, the whole beam will hit you and maximum efficiency will be achieved.

In non ideal situations the "mirror" will not be a sphere with your spacecraft as the center so the beam will not hit you 100%, also if the spacecraft is of a size similar to the wave diffraction will become significative and the photons as waves will travel in multiple directions, in complete chaos the light will hit you from all the sides and your spacecraft will flight like the relativistic rocket equation says

$\begingroup$Thanks for the answer! So to clarify, do you think that the proposed engine would work if the antenna were large enough and had a sufficiently large gain?$\endgroup$
– ThorondorAug 29 '18 at 3:09

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$\begingroup$i think it will work until the reflective surface (plasma) no longer forms a sphere around the spacecraft because of turbulence and convection$\endgroup$
– 8thsizecountryAug 31 '18 at 3:57